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Cohen-Macaulay Representations

Graham J. Leuschke Syracuse University, Syracuse, NY
Roger Wiegand University of Nebraska-Lincoln, Lincoln, NE
Available Formats:
Hardcover ISBN: 978-0-8218-7581-0
Product Code: SURV/181
367 pp
List Price: $102.00 MAA Member Price:$91.80
AMS Member Price: $81.60 Electronic ISBN: 978-0-8218-8790-5 Product Code: SURV/181.E 367 pp List Price:$96.00
MAA Member Price: $86.40 AMS Member Price:$76.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $153.00 MAA Member Price:$137.70
AMS Member Price: $122.40 Click above image for expanded view Cohen-Macaulay Representations Graham J. Leuschke Syracuse University, Syracuse, NY Roger Wiegand University of Nebraska-Lincoln, Lincoln, NE Available Formats:  Hardcover ISBN: 978-0-8218-7581-0 Product Code: SURV/181 367 pp  List Price:$102.00 MAA Member Price: $91.80 AMS Member Price:$81.60
 Electronic ISBN: 978-0-8218-8790-5 Product Code: SURV/181.E 367 pp
 List Price: $96.00 MAA Member Price:$86.40 AMS Member Price: $76.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$153.00
MAA Member Price: $137.70 AMS Member Price:$122.40
• Book Details

Mathematical Surveys and Monographs
Volume: 1812012
MSC: Primary 13; 16;

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.

Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3–10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material—ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures—is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Research mathematicians interested in algebra, in particular, theory of rings and modules.

• Chapters
• 1. The Krull-Remak-Schmidt theorem
• 2. Semigroups of modules
• 3. Dimension zero
• 4. Dimension one
• 5. Invariant theory
• 6. Kleinian singularities and finite CM type
• 7. Isolated singularities and dimension two
• 8. The double branched cover
• 9. Hypersurfaces with finite CM type
• 10. Ascent and descent
• 11. Auslander-Buchweitz theory
• 12. Totally reflexive modules
• 13. Auslander-Reiten theory
• 14. Countable Cohen-Macaulay type
• 15. The Brauer-Thrall conjectures
• 16. Finite CM type in higher dimensions
• 17. Bounded CM type
• Appendix A. Basics and background
• Appendix B. Ramification theory

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Volume: 1812012
MSC: Primary 13; 16;

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.

Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3–10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material—ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures—is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Research mathematicians interested in algebra, in particular, theory of rings and modules.

• Chapters
• 1. The Krull-Remak-Schmidt theorem
• 2. Semigroups of modules
• 3. Dimension zero
• 4. Dimension one
• 5. Invariant theory
• 6. Kleinian singularities and finite CM type
• 7. Isolated singularities and dimension two
• 8. The double branched cover
• 9. Hypersurfaces with finite CM type
• 10. Ascent and descent
• 11. Auslander-Buchweitz theory
• 12. Totally reflexive modules
• 13. Auslander-Reiten theory
• 14. Countable Cohen-Macaulay type
• 15. The Brauer-Thrall conjectures
• 16. Finite CM type in higher dimensions
• 17. Bounded CM type
• Appendix A. Basics and background
• Appendix B. Ramification theory
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